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19
Percolation in the Hyperbolic Plane
, 2000
"... Following is a study of percolation in the hyperbolic plane H 2 and on regular tilings in the hyperbolic plane. The processes discussed include Bernoulli site and bond percolation on planar hyperbolic graphs, invariant dependent percolations on such graphs, and PoissonVoronoiBernoulli percolation ..."
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Cited by 34 (1 self)
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Following is a study of percolation in the hyperbolic plane H 2 and on regular tilings in the hyperbolic plane. The processes discussed include Bernoulli site and bond percolation on planar hyperbolic graphs, invariant dependent percolations on such graphs, and PoissonVoronoiBernoulli percolation. We prove the existence of three distinct nonempty phases for the Bernoulli processes. In the first phase, p ∈ (0, pc], there are no unbounded clusters, but there is a unique infinite cluster for the dual process. In the second phase, p ∈ (pc, pu), there are infinitely many unbounded clusters for the process and for the dual process. In the third phase, p∈[pu, 1), there is a unique unbounded cluster, and all the clusters of the dual process are bounded. We also study the dependence of pc in the PoissonVoronoiBernoulli percolation process on the intensity of the underlying Poisson process.
UNIQUENESS FOR THE SIGNATURE OF A PATH OF BOUNDED VARIATION AND THE REDUCED PATH GROUP
, 2006
"... Abstract. We introduce the notions of treelike path and treelike equivalence between paths and prove that the latter is an equivalence relation for paths of finite length. We show that the equivalence classes form a group with some similarity to a free group, and that in each class there is one sp ..."
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Cited by 6 (1 self)
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Abstract. We introduce the notions of treelike path and treelike equivalence between paths and prove that the latter is an equivalence relation for paths of finite length. We show that the equivalence classes form a group with some similarity to a free group, and that in each class there is one special tree reduced path. The set of these paths is the Reduced Path Group. It is a continuous analogue to the group of reduced words. The signature of the path is a power series whose coefficients are definite iterated integrals of the path. We identify the paths with trivial signature as the treelike paths, and prove that two paths are in treelike equivalence if and only if they have the same signature. In this way, we extend Chen’s theorems on the uniqueness of the sequence of iterated integrals associated with a piecewise regular path to finite length paths and identify the appropriate extended meaning for reparameterisation in the general setting. It is suggestive to think of this result as a noncommutative analogue of the result that integrable functions on the circle are determined, up to Lebesgue null sets, by their Fourier coefficients. As a second theme we give quantitative versions of Chen’s theorem in the case of lattice paths and paths with continuous derivative, and as a corollary derive results on the triviality of exponential products in the tensor algebra. 1.
The number of unbounded components in the Poisson Boolean model of continuum percolation in hyperbolic space
, 2007
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CONTINUUM PERCOLATION AT AND ABOVE THE UNIQUENESS TRESHOLD ON HOMOGENEOUS SPACES
, 711
"... Abstract. We consider the Poisson Boolean model of continuum percolation on a homogeneous space M. Let λ be the intensity of the underlying Poisson process. Let λu be the infimum of the set of intensities that a.s. produce a unique unbounded component. First we show that if λ> λu then there is a. ..."
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Abstract. We consider the Poisson Boolean model of continuum percolation on a homogeneous space M. Let λ be the intensity of the underlying Poisson process. Let λu be the infimum of the set of intensities that a.s. produce a unique unbounded component. First we show that if λ> λu then there is a.s. a unique unbounded component at λ. Then we let M = H 2 × R and show that at λu there is a.s. not a unique unbounded component. These results are continuum analogies of theorems by Häggström, Peres and Schonmann. 1. Introduction and
DIMENSION REDUCTION FOR HYPERBOLIC SPACE ITAI BENJAMINI AND YURY MAKARYCHEV
"... Abstract. A dimension reduction for hyperbolic space is established. When points are far apart, an embedding with bounded distortion into H 2 is achieved. 1. ..."
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Abstract. A dimension reduction for hyperbolic space is established. When points are far apart, an embedding with bounded distortion into H 2 is achieved. 1.
1 The number of unbounded components in the PoissonBoolean model in H 2
, 2008
"... We consider the PoissonBoolean model with unit radius in the hyperbolic disc H 2. Let λ be the intensity of the underlying Poisson process, and let NC denote the number of unbounded components of the covered region. We show that there are two intensities λc and λu, 0 < λc < λu < ∞, such th ..."
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We consider the PoissonBoolean model with unit radius in the hyperbolic disc H 2. Let λ be the intensity of the underlying Poisson process, and let NC denote the number of unbounded components of the covered region. We show that there are two intensities λc and λu, 0 < λc < λu < ∞, such that NC = 0 for λ ∈ (0,λc], NC = ∞ for λ ∈ (λc,λu), and NC = 1 for λ ∈ [λu, ∞). Corresponding results, due to Benjamini, Lyons, Peres and Schramm, are available for Bernoulli bond and site percolation on certain nonamenable transitive graphs, and we use many of their techniques in our proofs. 1
Visualizing and Managing Cloud Computing Networks
"... Abstract—This paper describes the methods used to visualize and identify possible problems within a cloud computing network. A technique, rendering the entire cloud network in Hyperbolic space, for visualizing the physical layout of the network is discussed. Issues concerning scalability with regar ..."
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Abstract—This paper describes the methods used to visualize and identify possible problems within a cloud computing network. A technique, rendering the entire cloud network in Hyperbolic space, for visualizing the physical layout of the network is discussed. Issues concerning scalability with regards to effectively identifying problems within the cloud are discussed. The Sun Cloud Manager (Sun – Stellenbosch University) was created to demonstrate the techniques and data were generated to simulate realtime performance data obtained from the network. Index Terms—Cloud computing, visualization I.